Let ( E α ) α ∈ I , where I is an index set (for example, [0,1]). Can we define the corresponding liminf, for example, lim inf α ↘ 0 E α = ⋃ r &gt; 0 ⋂ 0 &lt; α &lt; r E α .My desired result is the following continuous parameter type Fatou's lemma: Let μ be a finite meaure and each E α is measurable, then μ ( lim inf α ↘ 0 E α ) ≤ lim inf α ↘ 0 μ ( E α ) .I am not sure whether these make sense. Any comments are welcome! Many thanks!

Question

Let   (      E    α        )          α      ∈      I      , where   I is an index set (for example, [0,1]). Can we define the corresponding liminf, for example,      lim inf          α      ↘      0            E    α    =      ⋃          r      &amp;gt;      0            ⋂          0      &amp;lt;      α      &amp;lt;      r            E    α    .My desired result is the following continuous parameter type Fatou&#039;s lemma: Let   μ be a finite meaure and each       E    α   is measurable, then  μ  (      lim inf          α      ↘      0            E    α    )  ≤      lim inf          α      ↘      0        μ  (      E    α    )  .I am not sure whether these make sense. Any comments are welcome! Many thanks!

Gornil2 · Accepted Answer

Note that                              ⋃                      r            &amp;gt;            0                                    ⋂                      0            &amp;lt;            α            &amp;lt;            r                                    E                      α                          =                  ⋃                      n            =            1                                ∞                                    ⋂                      0            &amp;lt;            α            &amp;lt;            1                          /                        n                                    E                      α                          .            Indeed, for   r  &amp;gt;  1, we have                              ⋂                      0            &amp;lt;            α            &amp;lt;            r                                    E                      α                          ⊆                  ⋂                      0            &amp;lt;            α            &amp;lt;            1                                    E                      α                          ⊆                  ⋃                      n            =            1                                ∞                                    ⋂                      0            &amp;lt;            α            &amp;lt;            1                          /                        n                                    E                      α                          .            Now we let                              S                      n                          =                  ⋂                      0            &amp;lt;            α            &amp;lt;            1                          /                        n                                    E                      α                          ,            one sees that   (      S          n            )          n      =      1              ∞       is an increasing sequence. Hence,                    μ        (                  lim inf                      α                                    E                      α                          )        =        μ                  (                      ⋃                          n              =              1                                      ∞                                            S                          n                                )                =                  lim                      n                          μ        (                  S                      n                          )        .            But then for a fixed   n, we have                    μ        (                  S                      n                          )        ≤        μ        (                  E                      α                          )        ,                0        &amp;lt;        α        &amp;lt;        1                  /                n        ,            this allows us to take lim inf both sides with respect to   α  ↓  0, which leads to                    μ        (                  S                      n                          )        ≤                  lim inf                      α                          μ        (                  E                      α                          )        .            Now we take   n  →  ∞ and note that       lim inf          α        μ  (      E          α        ) is a constant with respect to   n and we are done.

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